Properties

Label 2-1148-41.32-c1-0-21
Degree $2$
Conductor $1148$
Sign $-0.612 + 0.790i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.38 − 2.38i)3-s − 0.515i·5-s + (0.707 − 0.707i)7-s − 8.37i·9-s + (−3.73 + 3.73i)11-s + (0.0725 − 0.0725i)13-s + (−1.22 − 1.22i)15-s + (−4.03 − 4.03i)17-s + (−2.33 − 2.33i)19-s − 3.37i·21-s + 4.66·23-s + 4.73·25-s + (−12.8 − 12.8i)27-s + (6.22 − 6.22i)29-s − 2.93·31-s + ⋯
L(s)  = 1  + (1.37 − 1.37i)3-s − 0.230i·5-s + (0.267 − 0.267i)7-s − 2.79i·9-s + (−1.12 + 1.12i)11-s + (0.0201 − 0.0201i)13-s + (−0.317 − 0.317i)15-s + (−0.977 − 0.977i)17-s + (−0.536 − 0.536i)19-s − 0.736i·21-s + 0.972·23-s + 0.946·25-s + (−2.46 − 2.46i)27-s + (1.15 − 1.15i)29-s − 0.526·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.612 + 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.326591670\)
\(L(\frac12)\) \(\approx\) \(2.326591670\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-5.72 + 2.86i)T \)
good3 \( 1 + (-2.38 + 2.38i)T - 3iT^{2} \)
5 \( 1 + 0.515iT - 5T^{2} \)
11 \( 1 + (3.73 - 3.73i)T - 11iT^{2} \)
13 \( 1 + (-0.0725 + 0.0725i)T - 13iT^{2} \)
17 \( 1 + (4.03 + 4.03i)T + 17iT^{2} \)
19 \( 1 + (2.33 + 2.33i)T + 19iT^{2} \)
23 \( 1 - 4.66T + 23T^{2} \)
29 \( 1 + (-6.22 + 6.22i)T - 29iT^{2} \)
31 \( 1 + 2.93T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
43 \( 1 - 1.10iT - 43T^{2} \)
47 \( 1 + (-8.99 - 8.99i)T + 47iT^{2} \)
53 \( 1 + (-3.20 + 3.20i)T - 53iT^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 9.12iT - 61T^{2} \)
67 \( 1 + (-1.97 - 1.97i)T + 67iT^{2} \)
71 \( 1 + (-8.02 + 8.02i)T - 71iT^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + (-1.52 + 1.52i)T - 79iT^{2} \)
83 \( 1 - 5.12T + 83T^{2} \)
89 \( 1 + (11.3 - 11.3i)T - 89iT^{2} \)
97 \( 1 + (-10.0 - 10.0i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.133066783073013947648858182535, −8.677313785753576655361240189856, −7.78967160009919815928777308764, −7.12439223474528317921666865968, −6.68401962428144436294403919862, −5.14986248800809997293577709434, −4.14390751181902277703392105208, −2.68295618133095640711484683812, −2.29039788202114144803621563614, −0.840769367060948932839155094207, 2.15538202950188637275137213210, 3.03387786547572308905178295173, 3.77501647876095263502472547108, 4.84037277741377095160960653663, 5.51449797064079130602968179827, 6.93374890719917990412710617080, 8.079086522025405883043790981155, 8.704139885710838625182615148805, 8.909335569924245781243593476618, 10.28612535740816753711590514288

Graph of the $Z$-function along the critical line